Volatility Models: ARCH and GARCH
Why returns are unpredictable in the mean but not in the variance, how GARCH captures volatility clustering, and what its parameters imply.
Leads to: GARCH volatility feeds option pricing (Phase 15) and risk models throughout.
Learning Objectives
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- Explain volatility clustering and why it violates the constant-variance assumption of ARMA.
- Define the ARCH(\(q\)) and GARCH(\(p,q\)) models and interpret their parameters.
- Derive the stationarity condition and unconditional variance of GARCH(1,1).
- Relate GARCH persistence \(\alpha+\beta\) to volatility half-life and the IGARCH boundary.
- Simulate a GARCH process and detect clustering via the ACF of squared returns.
Key Vocabulary
- Volatility clustering
- The empirical tendency of large moves to follow large moves: \(|r_t|\) is autocorrelated even when \(r_t\) is not.
- Conditional variance
- \(\sigma_t^2=\Var(r_t\mid\mathcal F_{t-1})\), the variance given the past; time-varying in ARCH/GARCH.
- ARCH(\(q\)
- \(\sigma_t^2=\omega+\sum_{i=1}^q\alpha_i r_{t-i}^2\); conditional variance driven by recent squared returns.
- GARCH(\(p,q\)
- Adds lagged variances: \(\sigma_t^2=\omega+\sum\alpha_i r_{t-i}^2+\sum\beta_j\sigma_{t-j}^2\).
- Persistence
- \(\alpha+\beta\) in GARCH(1,1); how slowly volatility shocks decay; near 1 means very persistent.
- Unconditional variance
- \(\omega/(1-\alpha-\beta)\) for stationary GARCH(1,1); the long-run average variance.
- IGARCH
- The boundary \(\alpha+\beta=1\): shocks to variance never die out (integrated volatility).
Intuition & Motivation
ARCH makes today’s variance a function of recent squared shocks: a big surprise raises tomorrow’s expected variance. GARCH adds a smoothing term - today’s variance also remembers yesterday’s variance - which captures the long, gentle decay of real volatility with very few parameters. The single number \(\alpha+\beta\) summarizes how persistent turbulence is: close to 1 (as it usually is for equities) means a volatility shock takes many weeks to fade, and exactly 1 means it never does.
Why ARMA is not enough
Write a return as \(r_t=\sigma_t z_t\) with i.i.d. standardized \(z_t\) (\(\E z_t=0,\ \Var z_t=1\)) and a time-varying conditional standard deviation \(\sigma_t\). Then \(r_t\) is serially uncorrelated (so its ACF is flat, matching returns) but \(r_t^2\) is autocorrelated (matching the clustering). The mean model and the variance model are separate: ARMA handles the mean, ARCH/GARCH the conditional variance \(\sigma_t^2\).
ARCH and GARCH defined
with \(\omega\gt 0,\ \alpha_i,\beta_j\ge0\) to keep \(\sigma_t^2\gt 0\). GARCH(1,1) - just three parameters - is the workhorse and fits most return series remarkably well. Parameters are estimated by maximum likelihood: assuming \(z_t\sim\Normal(0,1)\), maximize \(\ell=-\tfrac12\sum_t\big(\log\sigma_t^2+r_t^2/\sigma_t^2\big)\) over \((\omega,\alpha,\beta)\) (a numerical optimization, since \(\sigma_t^2\) is recursive).
Persistence \(\alpha+\beta\) controls the decay of a volatility shock: the conditional variance mean-reverts toward \(\bar\sigma^2\) at rate \(\alpha+\beta\) per step, giving a half-life \(\log(1/2)/\log(\alpha+\beta)\). For daily equity returns \(\alpha+\beta\) is commonly \(\approx0.98\text{--}0.99\), a half-life of weeks. GARCH also generates fat tails in the unconditional return distribution even with Gaussian \(z_t\), matching a key stylized fact; a Student-\(t\) innovation fits the tails even better.
Interactive: simulate GARCH and see the clustering
- Applying ARMA to returns and assuming constant variance - it misses clustering entirely and mis-states risk.
- Reading GARCH as a mean-return forecast; it forecasts variance, not direction.
- Estimating GARCH without the constraints \(\omega\gt 0,\ \alpha,\beta\ge0,\ \alpha+\beta\lt 1\); violating them gives negative or explosive variances.
- Assuming Gaussian innovations when tails are heavy; use Student-\(t\) and check standardized-residual diagnostics.
- Ignoring leverage (down moves raise vol more than up moves); symmetric GARCH misses it - use EGARCH/GJR.
- Fit the mean (often just a constant, or an AR(1)) and the variance jointly, or fit GARCH to the mean residuals.
- Report persistence \(\alpha+\beta\) and the implied half-life; near-1 persistence warns that shocks fade slowly.
- Check standardized residuals \(r_t/\hat\sigma_t\): they should be i.i.d. with no remaining ACF in their squares.
- For asymmetry, prefer GJR-GARCH or EGARCH; for tails, Student-\(t\) innovations - both matter for equity risk.
Knowledge Check
Practical Exercise
A GARCH(1,1) fit to daily equity returns yields \(\omega=0.00001,\ \alpha=0.08,\ \beta=0.90\) (returns in decimals). (a) Is it stationary, and what is the unconditional daily volatility? (b) What is the persistence and the volatility half-life? (c) Interpret what happens to forecast volatility after a large market shock.
(a) \(\alpha+\beta=0.98\lt 1\), so it is covariance-stationary. Unconditional variance \(\bar\sigma^2=\omega/(1-\alpha-\beta)=0.00001/0.02=0.0005\), so daily volatility \(\bar\sigma=\sqrt{0.0005}\approx0.0224\), about 2.24% per day (roughly \(0.0224\sqrt{252}\approx35\%\) annualized).
(b) Persistence \(\alpha+\beta=0.98\). Half-life \(=\log(0.5)/\log(0.98)\approx34\) trading days: a volatility shock takes about a month and a half to halve.
(c) A large \(r_{t-1}^2\) spikes \(\sigma_t^2\) through the \(\alpha\) term; because \(\beta=0.90\) is large, the elevated variance then feeds forward and decays only slowly toward \(\bar\sigma^2\). So after a shock the model forecasts a persistent regime of elevated volatility that mean-reverts over roughly the half-life - capturing the clustering seen in crises.
Lesson Summary
Formula Sheet Additions
- Did I enforce \(\omega\gt 0,\ \alpha,\beta\ge0,\ \alpha+\beta\lt 1\) in estimation?
- Did I model the mean separately and fit GARCH to residuals?
- Are standardized residuals i.i.d. with no ACF in their squares?
- Did I consider fat-tailed (Student-\(t\)) innovations and leverage (EGARCH/GJR)?
Retrieval Practice
Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.
A: ARMA assumes constant variance but returns show volatility clustering (autocorrelated squares); ARCH/GARCH model the time-varying conditional variance \(\sigma_t^2\).
A: Covariance-stationary iff \(\alpha+\beta\lt 1\); unconditional variance \(\omega/(1-\alpha-\beta)\).
A: Volatility shocks decay very slowly (long half-life); at exactly 1 (IGARCH) they never die out.
Flashcards
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Completion Checklist
- I can explain the core ideas in my own words
- I worked the derivations/examples by hand
- I completed the interactive workbench(es)
- I passed the knowledge check